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GENERIC NUMERICAL CHALLENGES *
Robert Wainwright
Background:
Generic Numerical Challenges is based on an idea originally conceived over a century ago by
William Rouse Ball [1]. Martin Gardner popularized this by challenging his readers to solve a
set of problems in “The Numerology of Dr. Matrix” [2]. In 2008, the author introduced this
concept as Digital Challenges and later in 2010 as Numerical Challenges to the Gardner
Gathering audience [3], [4]. In 2012, a more recent reference to this idea appeared [5].
The general objective of these challenges is to develop mathematical expressions which equal a
given target number using a given set of base digits selected from the ten integers (0 though 9).
Allowed mathematical operations may be either elementary, which include addition [+],
subtraction [-], multiplication [x], and division [/]; intermediate, which also allows decimals [.];
or advanced, which allows other variations such as exponents [^], roots [√], factorials [!],
repeating digits [.nr] or concatenation [nn].
Some examples of these for a target of ten using the base digit set of one, two, three, and four are
shown here:
Elementary Operations:
10 = 1 +2 + 3 + 4
10 = (1)(3)(4) – 2
Intermediate Operations:
10 = (2 + 3 - 4) / .1
10 = (2 / .4)(3 - 1)
Advanced Operations:
10 = 3√4] + 12
10 = 3!! / ((1 + 2)(4!))
Concept of Generic Digits:
It is interesting to note that some of the base digits need not be exactly specified, that is, they
may be unknown in value. We will call these generic digits designated by the letter g.
Interestingly, if we have a pair of unknown but equal generic digits, many precise values may be
determined.
For example:
g - g = 0 and g / g = 1
_____________________________________________________________________________
* Presented at the Tenth Gathering for Gardner, Atlanta GA, March 2012.
1.
By allowing decimals:
.g / g = 1/10 or .1 and g / .g = 10
By allowing repeating decimals:
.gr / g = 1/9 or .111... (i.e. .1r) and g / .gr = 9
.g / .gr = 9/10 and .gr / .g = 10/9 or 1.111... (i.e. 1.1r)
If roots are allowed:
√.gr/g] = 1/3 or .333... (i.e. .3r) and √g/.gr] = 3
Also, using factorials:
(g - g)! = 1 and √g/.gr]! = 6 and √g/.gr]!! = 720 and so forth.
This generic concept forms the basis for several challenges by using letters and numbers in the
Gardner Gathering logo(s), G4G10 and G4GX.
Specific Challenges:
Listed below are several challenges all requiring a set of base digits taken from the symbols
found in the Gardner Gathering logo. All except the last of these asks for solutions to produce a
particular target number. In every case, use of advanced operations is allowed.
* Target = 5 using base set = 4, 10, g, g (seven different solutions known).
* Target = 7 "
"
" = 0, 1, 4, g, g (sixteen
"
"
"
).
* Target = 10 "
"
" = 0, 1, 4, g, g (thirteen "
"
"
).
* Target = 47 "
"
" = 0, 1, 4, g, g (seven
"
"
).
"
* Target (X = 0 ... 15) using base set = 4, g, g forms the basis for exchange "Logo Puzzle".
Related Follow up:
Math DiceTM, invented by Sam Ritchie and produced by Thinkfun ©, consists of problems using
ordinary dice which are thrown to produce both a target number and also a "scoring number"
(base set of numerical digits). In this game the player is asked to use math skills to come closest
to the given target number.
Finally, more than two of the base digits could be generic. It is interesting to note that with three
generic digits (g, g, and g), forty other values including 2, 5, 8, and 11 are possible.
2.
References
[1] W. W. Rouse Ball, Mathematical Recreations and Essays, Macmillan and Co., NY, 1892.
[2] Martin Gardner, The Numerology of Dr. Matrix, Simon and Schuster, New York , 1967.
[3] R. Wainwright, Eighth Gathering for Gardner, Atlanta, 2008.
[4] R. Wainwright, Ninth Gathering for Gardner, Atlanta, 2010.
[5] R. Wainwright, A Numerical Challenge, College Math J., 43, 1; (2012) 19.
For further information including solutions to the Challenges presented, please contact:
Robert Wainwright
12 Longue Vue Avenue
New Rochelle NY 10804
[email protected]
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g4gX 21mar12